a result, thermal convection occurs as thermal columns carry hot material to the surface (photosphere) of the Sun. Once the material cools off at the surface, it plunges back downward to the base of the convection zone, to receive more heat from the top of the radiative zone. In a solar model, as described in stellar structure, one considers the density , temperature T(r), total pressure (matter plus radiation) P(r), luminosity l(r) and energy generation rate per unit mass ε(r) in a spherical shell of a thickness dr at a distance r from the center of the star. Radiative transport of energy is described by the radiative temperature gradient equation: where κ is the opacity of the matter, σ is the Stefan–Boltzmann constant, and the Boltzmann constant is set to one. Convection is described using mixing length theory and the corresponding temperature gradient equation (for adiabatic convection) is: where is the adiabatic index, the ratio of specific heats in the gas. (For a fully ionized ideal gas, .) Near the base of the Sun's convection zone, the convection is adiabatic, but near the surface of the Sun, convection is not adiabatic. Simulations of near-surface convection A more realistic description of the uppermost part of the convection zone is possible through detailed three-dimensional and time-dependent hydrodynamical simulations, taking into account radiative transfer in the atmosphere. Such simulations successfully reproduce the observed surface structure of solar granulation, as well as detailed profiles of lines in the solar radiative spectrum, without the use of parametrized models of turbulence. The simulations only cover a very small fraction of the solar radius, and are evidently far too time-consuming to be included in general solar modeling. Extrapolation of an averaged simulation through the adiabatic part of the convection zone by means of a model based on the mixing-length description, demonstrated that the adiabat predicted by the simulation was essentially consistent with the depth of the solar convection zone as determined from helioseismology. An extension of mixing-length theory, including effects of turbulent pressure and kinetic energy, based on numerical simulations of near-surface convection, has been developed. This section is adapted from the Christensen-Dalsgaard review of helioseismology, Chapter IV. Equations of state The numerical solution of the differential equations of stellar structure requires equations of state for the pressure, opacity and energy generation rate, as described in stellar structure, which relate these variables to the density, temperature and